The application of this idea to the algebro-geometric method
can be found in the wonderful works [1]}. In [2]}
we have constructed the algebro-geometric solutions of FL hierarchy
using the algebro-geometric method [3]}.
The basic strategy we shall take
is to
apply the idea
of
[1]}
to [2]} and
the
technique
developed
in this text
seems
be available
to solve the reality problems of
the algebro-geometric solutions
of
other integrable equations
via the algebro-geometric method [3]}.
Here we should emphasize
the work of the authors in [3]}.
They
have discussed the reduction conditions for the
algebro-geometric solutions of
AKNS hierarchy using Vinikov's basis (for the definition, see Proposition 1.1.1).
However, it may
be very complicated to depict the Vinikov's basis for a general real curve,
although this problem become simple in some special cases (e.g. all the
branch points of the spectral curve are real). Moreover,
various of path integration in explicit
algebro-geometric solutions
and the action of holomorphic involution on spectral curve
very often involves the specific path,
which inevitably connects with
the choice of homology basis. Thus, we shall
solve the reality problem by using very specific
curve and homology basis
and every parameter
in the finally
expressions
of solutions is very clear.
This is also
the biggest difference
from 'abstract' Vinikov's basis.
